(2) RIMS Kôkyûroku Bessatsu B31 (2012), 157166. Gray‐Scott model as a tool of thermodynamic investigation in Non‐Equilibrium Chemical Systems Reversible. By. Hitoshi Mahara. *. **. and Tomohiko Yamaguchi. Abstract In this report,. introduce the reversible. Gray‐Scott. model. tool of. thermodynamic non‐equilibrium investigation original Gray‐Scott model in order to calculate the entropy production of a reaction‐diffusion system. Here, we study the relations between the pattern formation and the thermodynamic quantities; the entropy production, the entropy flow and the entropy change. In summary, we comment the relation between the thermodynamic quantity and the mathematical analysis of pattern dynamics. we. reaction‐diffusion system.. in. §1.. ordered arrays of structures. seem. succeeded to. stripes, dots. explain. are. or. this contradiction and. evoked. April 12,. For. both appear in the skin patterns of fish law.. proposed. a. through only dissipative. 2011. Revised November. 1,. example,. [1].. These. Prigogine. et al.. concept of the dissipative. struc‐. thermodynamic. non‐equilibrium thermodynamics [2, 3].. librium. Therefore these ordered structures Received. especially biological system.. to contradict with the second. tures in the framework of the structures. Introduction. ordered structures exist in nature,. Many. as a. This model is modified from the. The. dissipative. processes under condition far from. satisfy. the second law of. equi‐. thermodynamics.. 2011.. Subject Classication(s): Subject Classication(s): Key Words: entropy balance: entropy produciton: entropy flow: reaction‐diffusion system: stable manifolds Supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) via Grant‐in‐Aids for Scientic Research on Innovative Areas Emergence in Chemistry. 2000 Mathematics. 2000 Mathematics. . *. (20111007). AIST,. Tsukuba Central 5,. Higashi, 1‐1‐1,Tsukuba, Ibaraki, 305‐8565, Japan. h‐mahara@aist.go.jp AIST, Tsukuba Central 5, Higashi, 1‐1‐1,Tsukuba, Ibaraki, 305‐8565, Japan. \mathrm{e} ‐mail: tomo.yamaguchi@aist.go.jp \mathrm{e} ‐mail:. **. ©. 2012 Research Institute for Mathematical. Sciences, Kyoto University.. All. rights. reserved..

(3) Hitoshi Mahara. 158. After the. equilibrium. proposal of dissipative. and. and. Tomohiko Yamaguchi. structures. dissipative systems from. the. concept, the number of studies in increased.. Especially. experimentally. in reaction. dynamical viewpoint. pattern dynamics. has been well studied. mathematically. diffusion systems. [4, 5, 6, 7]. However,. the number of those studies from the thermo‐. dynamic viewpoint between the. decreased.. Thus there. thermodynamic quantities. are. a. and. non‐. few studies that discuss the relation. and the pattern. dynamics. behavior of the. or. non‐equilibrium system. In order to discuss that. [8].. In this report,. we. present. relation, our. we. introduced the reversible. The relations between the pattern. tities. discussed.. In summary,. thermodynamic quantities. and the mathematical. the chemical reactions. tions of non‐linear pattern formation.. production of nite.. dynamics. and the. to calculate the. In this paper,. System. are. irreversible when these. However, we use. thermodynamic quantities. This model is called the reversible. original Gray‐Scott. model. [4].. This. quan‐. description of pattern dynamics.. it is. impossible. the irreversible reactions because the entropy. (See Appendix A). thermodynamic. add comments about the relation between the. we. §2.. Generally,. model. studies with the numerical calculation of the entropy. balance. are. Gray‐Scott. in. the reversible a. original. to calculate the. production. Gray‐Scott. reaction diffusion. Gray‐Scott model, model. was. used for simula‐. are. entropy. becomes infi‐. model in order. system.. which is modified from the considered. as. a. part of the. Brullseter model. The model consists of two chemical reaction steps and three chemical. spices U, W and. P:. (2.1). \mathrm{U}+2\mathrm{W}\Leftrightarrow 3\mathrm{W},. (2.2). \mathrm{W}\Leftrightarrow \mathrm{P}. The. (2.3). equations that. govern this reaction‐diffusion. system. are as. follows: Eq. 2.1. \left{bginary}{l \fracptilU}{\partil}=-UW^{2+f(1-U)k_{r}W^3+D_{U}\nabl^{2}U\ frac{\ptilW}{\partil}=UW^{2-(f+k)W_{r}^3+k_{r}PDW\nabl^{2}W\ frac{\ptilP}{\partil}=kW-_{rPf+D_{}\nabl^{2}P \end{ary}\ight..

(4) Reversible GRAY‐Scott. is the flow rate constant; k and. f. where. and the backward. of. \mathrm{U}, \mathrm{W} and. \mathrm{P} ,. k_{r}. are. the rate constants of the forward. reactions, respectively. D_{U} DWand D_{P} ,. and. respectively,. 159. model as a tool 0F thermodynamic investigation. are. the diffusion coefficients. are. D_{U}=2.0\times 10^{-5}, D_{W}=1.0\times 10^{-5}. set to be:. and. D_{P}=1.0\times 10^{-6}. Results and discussion. §3. The present system. quantities (entropy production, entropy flow and entropy change). in the two. that. cases. a. one‐dimensional media shows. The relations among. gating pulse.. flow,. we. introduced. The definition and calculation method of this. First,. (Fig.1).. we. The. show the results in the side of the system is. one. pulse self‐replicates after is filled with four. The entropy. a. pulses. Finally, production. Finally,. pulse a. entropy production is proportional. production. is. an. index of pattern. production. cannot work. a. In the. as a. structures. negative. shows. this. near zero. quantity. previous. converges to zero,. change. mean. as. and it. the system shows. quasi‐. regions,. paper. potential function. [8],. state.. The. and then the entropy we. examined whether. and it is. the result shows. counterexample. a. occurs. because of the. [8].. values.. This. means. that the system throws out. The absolute value of this. production.. It. means. quantity. is. that most amount of the. out to its environment in order to. change. is smaller than the other. value when the system shows. shows. self‐replicating. is. [11].. The absolute value of the entropy. quantity. pulse. system shows stable. of black. entropy that is produced inside the system is thrown. keep the system. B.. three times and the system. pulse replication phenomenon. that of the entropy. same. while the. area. the entropy to its environment with the flow. almost the. a. self‐replicating, i.e.,. is not. entropy production principle. The entropy flow shows. Appendix. self‐replicating pulse pulse is initiated. This. potential function. Unfortunately,. the entropy minimum. rapidly. [8, 9].. could be. that. Appendix. potential [11, 12].. chemical. and. pulse self‐replicates. to the. production. hypothesis. propa‐. that the system shows. constant when the. the entropy. of the Hansons. a. the system becomes stable.. increases. it becomes. potential. and. described in. are. is described in. perturbed initially. while. This. becomes almost stable while the. stable state.. case. a new. calculated. are. self‐replicating pulse. thermodynamic quantities. A. In order to calculate the entropy. hands,. self‐replicating pulse,. co‐existing patterns [9, 10]. Here, thermody‐. chaotic pattern and also. stretching line, namic. show various pattern formations:. can. a. peak while the pulse. is. quasi‐stable. quantities. This. state. On the other. self‐replicating.. This. quantity. the system becomes stable state. These behaviors of the entropy. that the entropy increases. during. the process of the. pulse of self‐replication. and the entropy becomes stable while the system shows static stable state.. Second, thermodynamic quantities. are. calculated in the. case. the system shows. a.

(5) Hitoshi Mahara. 160. and. Tomohiko Yamaguchi. \displayte\frc{math i}\mathr{i}S_\smI}{athrm}\in \displayte\frac{prtialS_{1'}\mathr{}_f. \displayte\frac{mthrf}]S{\mathri}]$\ota \lceil\mathrm{J}. \supset-[\rfloor[]1|. ][\displaystyle \rfloor[][|[] ]^{\underline{ $\sigma$}_{\backslash }}.[]\langle\int[]. time. Figure. 1.. Time series of the one‐dimensional. when the system shows. self‐replicating pulse.. system and thermodynamic quantities The system size is 180dx.. dx=0.005.. f=0.03, k=0.06, k_{r}= 0:001. The system has the Neumann boundary condition. First panel: time series of the system. The gray‐scale shows the concentration of \mathrm{U} 0.0. (black). panel:. to. 1.0(white).. time series of the. Second. panel:. time series of the. entropy production. Third. entropy flow. Forth panel: time series of the entropy change..

(6) Reversible GRAY‐Scott. 161. model as a tool 0F thermodynamic investigation. time. \displayte\frc{mathi}\_{mathrL}\mathr{p}_J\mathr{i}_ \displayte\frac{mthr{i}\mathr{i}S_1'\mathr{}_I. \displayte\frac{ptialS}{\mthr{a}_f []. \underline{7}[][][]. 4. \mathrm{r}1(\}[][]. \mathrm{R}. $\iota$. |\underline{7}[][]\mathrm{l}\}. time. Figure. 2.. Time series of the one‐dimensional. when the system shows k=. 0:047, k_{r}=. Neumann. The. boundary. gray‐scale panel:. The. to the. The system size is 400dx. dx=0.005.. boundary. periodic. condition is. one.. First. shows the concentration of U0.0. time series of the. Forth. 0:001.. traveling pulse.. system and thermodynamic quantities. entropy production.. time series of the. Third. entropy change.. changed. panel:. f=0.01,. 3000:0, from the. The time series of the system.. (black). panel:. at t=. to 1.0. (white).. time series of the. Second. panel:. entropy flow..

(7) Hitoshi Mahara. 162. and. Tomohiko Yamaguchi. traveling pulse (Fig.2). The system is perturbed and a pulse is initiated initially. After a while, the Neumann boundary is changed to the periodic boundary. Then the pulse is. with. propagating. a. constant values after the a. production. The entropy flow goes to. of the entropy. produced. then the entropy of the system is From these two cases, the. when the system shows. pattern is static. system. constant value that is the. always kept. it shows constant value when the. with constant. velocity.. that. becomes. Especially. whether the. zero. is. a. state function and then. Summary. introduce. we. means. The latter property coincides. i.e., the entropy. system with the reversible Gray‐Scott model.. change,. It. become constant.. change. calculate the entropy balance. we. zero.. system is stable.. §4.. and the entropy. converges to. absolute value. constant.. the entropy. with the intrinsic property of the entropy,. In this report,. change. same. inside the system is thrown out to the system and. solution,. or moves. entropy production becomes. kept constantly and the entropy. is. thermodynamic quantities. stable. a. become. region [12].. negative. a. Then the entropy. production.. all of the entropy that is. traveling pulse. to the black. proportional. is. of. thermodynamic quantities. is stabilized. The. traveling pulse profile. constant because the. All of the. velocity.. constant. a new. equation. in. a. reaction‐diffusion. In order to calculate the. chemical. potential.. entropy flow. The property of the. entropy change consists with the instinctive and intrinsic property of the entropy that the entropy is. state function of the. a. is useful to calculate. potential. Here, think. mathematically. back. the. on. numerically study. norm. case. the. stable,. These manifolds called. ( k is the number of the. the saddle node bifurcation. pulses)[6].. The. point of. moving point stays temporally. in other. moving. points of these branches in. words,. while the one. pulses. of the. near. this. point while the pulses. look stable.. pulses begin. to. However,. the. self‐replicate.. are. shows another. quasi‐stable. point settles down no. calculations,. stable. on. state. (looks. like k+1. Then the. pulses solution). Finally,. the stable manifold when the system shows. manifold,. the entropy. the system shows chaotic behaviors.. change. converges to zero, but not. zero. quasi‐. moving point gets. converges to the saddle node bifurcation of the k+1 mode branch and the. there is. Turing. self‐replicating pulses. The moving point converges to k‐mode branch when the k pulses exist in the system. After. away from this point when. point. equation.. that the system shows. the bifurcation. that,. pulses. near. chemical. Ueyama [6, 7]. They analyzed self‐replicating pulse. Their system has many. and parameter space.. branch of k mode for stable k. point of the system travels. the entropy balance. new. of Nishiura and. the mechanism of the. stable manifolds in the. the. system. Therefore, introducing the. a. the. moving system. moving. stable state.. If. From the present. while the. pulse. does.

(8) Reversible GRAY‐Scott. not. self‐replicate, i.e.,. up from. zero. the system shows. when the. pulse begins. to. change Turing. branch and then there is. Therefore,. relation between the. we. a. quasi‐stable. moving point. possibility. state. And also this. quantity. goes. Therefore it looks that the entropy. self‐replicate.. shows the distance between the. that distance.. 163. model as a tool 0F thermodynamic investigation. and the stable manifold of k mode. that the entropy. change. can. be. an. index of. believe that this relation is useful to reveal the detailed. thermodynamic quantities. and the mathematical. description of the. system.. References. [1] Kondo, [2] [3] [4] [5] [6] [7] [8]. Asai, R., A viable reaction‐diffusion wave on the skin of a marine angelfish Poacanthus, Nature, 376 (1995), 765768. Nicolis, G. and Prigogine, I., Self‐ Organization in Non‐Equilibrium System, Wiley & Sons, Inc,1977 Kondepudi, D., Introduction to Modern Thermodynamics, John Wiley & Sons, Ltd, 1998. Peason, J. E., Complex Patterns in a Simple System, Science, 261 (1993), 189192. De kepper, P., Castes, V., Dulos, E. and Boissonade, J., Turing‐type chemical pattern in chlorie‐iodide‐malonic acid reaction, Physica D, 49 (1991), 161169. Nishiura, Y. and Ueyama, D., A skeleton structure of self‐replicating dynamics, Physica D, 130 (1999), 73104. Nishiura, Y. and Ueyama, D., Spatio‐temporal chaos for the Gray‐Scott model, Physica D, 150 (2001), 137162. Mahara, H., Suematsu, N. J., Yamaguchi, T., Ohgane, K., Nishiura, N. and Shimo‐ mura, M., Three‐variable reversible Gray‐Scott model, J. Chem. Phys., 121 (2004), 8969‐ S. and. 8972.. [9] Mahara, H., Yamaguchi,. T. and. Shimomura, M., Entropy production in two‐dimensional reversible Gray‐Scott model, CHAOS, 15 (2005), 047508. [10] Mahara, H., Suzuki, K., Jahan, R. A, and Yamaguchi, T., Co‐existing stable patterns in a reaction‐diffusion system with reversible Gray‐Scott dynamics, Phys. Rev. E, 78 (2008), 066210.. [11] Mahara,. H. and. Yamaguchi, T., Entropy balance in distributed reversible Gray‐Scott model, Physica D, 239 (2010), 729734. [12] Mahara, H. and Yamaguchi, T., Calculation of entropy balance equation in a non‐ equilibrium reaction‐diffusion system, Entropy, 12 (2010), 24362449.. § Appendix The entropy balance namics.. This. equation. A.. equation. Entropy balance equation is. a. basic. is derived from the. between the entropy per unit volume and. law with chemical reactions is described. (A. 1). equation for non‐equilibrium thermody‐ mass. mass. conservation law and the relation. density [2, 3].. as. \displaystyle\frac{\partial$\rho$_{j}{\partialt}=-\mathrm{d}\mathrm{i}\mathrm{v}j_{}+\sum_{i}v_{ji}w_{i}. The. mass. conservation.

(9) Hitoshi Mahara. 164. where $\rho$_{j} and. j_{j}. are. density. v_{ji} is stoichiometric constant of. and. Tomohiko Yamaguchi. and flux vector of the. jth. chemical. species. jth. chemical. species, respectively.. in ith chemical reaction. w_{i} is the. rate of the ith chemical reaction.. The relation between the entropy per unit volume and. mass. density. is described. as. \displayst le\frac{\parti ls}{\parti lt}=-\sum_{j}\frac{$\mu$_{j} T\frac{\parti l$\rho$_{j} \parti lt}. (A.2) where $\mu$_{j} is the chemical. potential of. the. jth. chemical. species and. it is described. as:. (A.3). $\mu$_{j}=$\mu$_{j}^{*}+k_{B}T\ln c_{j}. where chemical. $\mu$_{j}^{*}. is the standard chemical. potential. and c_{j} is the concentration of the. species [2, 3]. The standard chemical potential has. teristic for each chemical. The entropy balance. a. jth. constant value charac‐. species. equation. consists of the three. thermodynamic quantities:. the. entropy change, the entropy production and the entropy flow. The entropy change is the time derivative of the entropy of the system, @S = @t, and the. equation. is described. as. \displaystyle\frac{\partialS}{\partialt}=\frac{\partialS_{i}{\partialt}+\frac{\partialS_{e}{\partialt}=\int_{V}($\sigma$- divJ). (A.4) where. @S = @t. =\displaystyle \int_{V} $\sigma$ dV. is the. dV. entropy production and @S = @t. =-\displaystyle \int_{V} divJdV. is. the entropy flow. The entropy. production represents. duced inside the system. The entropy scribed. the time derivative of the entropy that is pro‐. production of. a. reaction‐diffusion system is de‐. as:. (A.5). \displaystyle\frac{\partialS_{i}{\partialt}=\int_{V}$\sigma$dV. =\displaystyle \int_{V}\sum_{i}k_{B}(v_{i,+}-v_{i,-})\ln\frac{v_{i,+} {v_{i,-} dV+\int_{V}\sum_{j}k_{B}\frac{D_{j} {c_{j} (\nabla c_{j})^{2}dV. where k_{B} is the Boltzman constant; v_{i,+} and v_{i,-}. are. the rates of the forward and. the backward reactions of the ith chemical. coefficient and the concentration. reaction, respectively. D_{j} of the jth chemical species.. The entropy flow represents the time derivative of the entropy that goes out to the environment. be written. as an. integral. across. over. the. boundary. is the diffusion. comes. from. of the system. The entropy flow. the system surface $\Omega$ with the. divergence. theorem:. or. can.

(10) Reversible GRAY‐Scott. \displaystyle\frac{\partialS_{e}{\partialt}=-\int_{V}. (A.6). divJdV. =\displaystyle\int_{$\Omega$}\sum_{j}\frac{$\mu$_{j}{T}j_{}\cdotnd$\Omega$. where. n. is the unit vector normal. For the calculation of the. Eq.A.6. the system surface.. on. Calculation of the entropy flow. B.. § Appendix. $\mu$ in. 165. model as a tool 0F thermodynamic investigation. entropy flow, the flow. should be described. vector. j. and the chemical. concretely. Here, descriptions. potential. of these quantities in the. present model will be given. The flux vectors. are. derived from the system. configurations. Here, we consider a simplicity [11]. Spatial inhomogeneity in. one‐dimensional reaction‐diffusion system for concentrations appears. along. Inward flows of chemical. species. (B.1). direction. one. are. given. (B.2). are. given. the flow vector. j. is. described;. as:. as:. j_{U}=fUe_{z}, j_{W}=-fWe_{z}, j_{P}=fPe_{z},. where e_{z} is the unit vector directed the direction of this vector is set to. zero. boundary. at the Neumann. boundary. the standard chemical. impossible. inconvenience,. relative chemical. to the direction. conditions out. or. [11].. x. .. potentials. introduce. an. are. we. When the system has. unknown in. face our. a. problem. model.. potential,. This fact. periodic. x.. potential. is defined. species. means. that. To avoid this. $\mu$_{r} , which is called the. potential [11].. The relative chemical. are. that the values of. entropy flow from eqs.A.3 and A.6.. alternative chemical. Here,. The flow terms. in flow term in the direction. potential. However,. to calculate the. we. no. to the surface normal vector.. is described with the concentrations of the chemical. potential. and its standard chemical. parallel. perpendicular. condition the system has. The chemical. (B.3). only. Then,. j_{U}=-fe_{z}, j_{W}=0, j_{P}=0,. and the outbound flows. it is. x. as:. $\mu$_{r,j}=$\mu$_{j}^{*}+k_{B}T\displaystyle \ln c_{j}-$\mu$_{j}^{*}-k_{B}T\ln c_{e,j}=k_{B}T\ln\frac{c_{j} {c_{e,j}.

(11) Hitoshi Mahara. 166. and. where c_{e,j} is the concentration of the This. equilibrium. state is defined. should reach if the system is. as. the. suddenly. Tomohiko Yamaguchi. chemical. jth. steady. equations. state. are. c_{e,j}(=c_{e,j}(t)). obtained. at the. state to which the. potentials. that the. following three equations. system is isolated, i.e., f=0.0.. \displaystyle \int_{V}(U+W+P)dV=(U_{e}+W_{e}+P_{e})V,. (B.5). U_{e}W_{e}^{2}=k_{r}W_{e}^{3},. (B.6). kW_{e}=k_{r}P_{e},. where V is the volume of the system;. species \mathrm{U}, \mathrm{W} and. total number of molecules is and third to the. equations. mean. \mathrm{P} ,. preserved. U_{e}, W_{e} and P_{e}. respectively.. are. the. in the reversible. principle of. the. The first. equilibrium. equation. Gray‐Scott. that each chemical reaction step. equilibrium states, i.e.,. system, the. is calculated from the. by assuming. state.. non‐equilibrium system. in the present. (B.4). tions of chemical. equilibrium. isolated from its environment.. For the calculation of the relative chemical. equilibrium. species. (eqs.2.1. These. concentra‐. means. that the. model. The second and. 2.2). should go. the detailed balance should be satisfied..

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